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Energy-harvesting sliding-window constrained block codes guarantee that within any prescribed window of ℓ consecutive bits the constrained sequence has at least t, t≥1, 1’s. Prior art code design methods build upon the finite-state machine description of the (ℓ,t) constraint, but as the number of states equals ℓ choose t, a code design becomes prohibitively complex for mounting ℓ and t. We present a new block code construction that circumvents the enumeration of codewords using a finite-state description of the (ℓ,t)-constraint. The codewords of the block code are encoded and decoded using a single look-up table. For (ℓ=4,t=2), the new block codes are maximal, that is, they have the largest possible number of codewords for its parameters.
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Block Codes for Energy-Harvesting
Sliding-Window Constrained Channels
Kees A. Schouhamer Immink, Fellow, IEEE, and Kui Cai, Senior Member, IEEE
Abstract—Energy-harvesting sliding-window constrained block
codes guarantee that within any prescribed window of `con-
secutive bits the constrained sequence has at least t,t1, 1’s.
Prior art code design methods build upon the finite-state machine
description of the (`, t)constraint, but as the number of states
equals `choose t, a code design becomes prohibitively complex
for mounting `and t. We present a new block code construction
that circumvents the enumeration of codewords using a finite-
state description of the (`, t)-constraint. The codewords of the
block code are encoded and decoded using a single look-up table.
For (`= 4, t = 2), the new block codes are maximal, that is, they
have the largest possible number of codewords for its parameters.
Keywordsenergy harvesting sliding-window constrained
code, constrained code, code design, binary block code, code
construction.
I. INTRODUCTION
Not all electronic devices connected to the Internet of
Things (IoT) have back-up batteries or are connected to the
power cable. These self-contained devices must rely on the
harvesting of the signal’s energy sent by a transmitter [1, 2].
Binary data communication systems may emit 0’s and 1’s,
where it is assumed that only the 1’s carry the energy. In
order to carry sufficient energy, a minimal and/or maximal
number of 1’s in transmitted sequences is required within a
prescribed time span. In the wireless infrared channel, IrDA-
VFlr, the relative number of 1’s, called duty cycle, must be
bounded between a minimum and maximum number [3]. A
binary sequence is said to obey the (`, t)-constraint if the
number of 1’s within any window of `consecutive bits of
that sequence is at least t,t < `, where tis an integer called
threshold. The (`, t)constraint is described by a finite-state
machine (FSM) source with K=`
tstates and a K×K
transition matrix D`,t [4, 5, 6].
An (`, t)constrained code translates a series of m-bit user
words into a series of n-bit codewords, n > m, that obeys
the (`, t)constraint. Prior art systematic code design methods,
such as Freiman and Wyner’s optimum block code design [7],
Franaszek’s successive state elimination procedure [8], the
state-splitting procedure [9], or the variable-length constrained
code construction [2, 10, 11], use the n-step FSM description
for enumerating codewords.
Kees A. Schouhamer Immink is with Turing Machines Inc, Willem-
skade 15d, 3016 DK Rotterdam, The Netherlands. E-mail: immink@turing-
machines.com.
Kui Cai is with Singapore University of Technology and Design (SUTD),
Science, Mathematics and Technology Cluster, 8 Somapah Rd, 487372,
Singapore. E-mail: cai kui@sutd.edu.sg.
This work is supported by Singapore Ministry of Education Academic
Research Fund Tier 2 MOE2016-T2-2-054.
These systematic design methods, as outlined in [6], run out
of steam for larger values of `and tas the crucial K×K
n-step transition matrix Dn
`,t becomes too large to handle.
For example, for (`= 30, t = 15) the K-th order matrix,
Dn
`,t, requires around 2×1016 elements, and, evidently, a
Dn
`,t-based code design is impractical for mounting `and t.
It is a desideratum to design (`, t)-constrained codes with
larger values of `and tthan is possible with the prior art
constructions. In this paper, we present an alternative code
design that circumvents the usage of an FSM description of
the (`, t)-constraint, so avoiding the computation of the n-step
transition matrix Dn
`,t.
The main contributions of the paper are as follows.
We present a general design method for constructing (`, t)-
constrained block codes.
The design method does not require the n-step transition
matrix Dn
`,t, so that we may construct codes for large values
of `and t, where prior art construction methods cannot be
used.
Examples of new codes are presented.
We assess the rate efficiency of codes based on the new
construction, and show that (4,2)-constrained block codes are
maximal, that is, they have the largest possible number of
codewords for its parameters.
The paper is organized as follows. We describe definitions
and background in Section II, followed by Section III, where
we present a new method for constructing block codes. The
efficiency of the codes constructed by the new method is
assessed in Sections IV and V. Section VI furnishes the
conclusions of our paper.
II. DE FIN IT IONS AND BACKGRO UN D
Let xbe an `-bit word x= (x1, x2, . . . , x`)over the
binary symbol alphabet xiB,B={0,1}, and let the
weight w(x) = P`
i=1 xibe the number of 1’s in x. Let
(y1, y2, . . . , yn)be a binary sequence of length n,n`. A
sequence (y1, y2, . . . , yn)is said to be sliding-window energy
(`, t)-constrained if for all 1in`+ 1 the weight
of the `-bit sliding window, w(yi,...yi+`1)t, where t,
1t`, is a positive integer.
The (`, t)-constrained channel can be modelled by a K-state
source [6], K=`
t, whose states are denoted by σiΣ,
1iK, where Σdenotes the state set. The states are
uniquely labelled with `-bit words of weight t, which are
assigned according to their lexicographical rank to the K
states, σ1, . . . , σK. The K×Ktransition matrix D`,t whose
elements di,j Brepresent a possible transition from state
σito state σj. If a transition is admissible then di,j = 1 and
σ1
0011
σ2
0101
σ3
0110
σ4
1001
σ5
1010
σ6
1100
1
0
1
01
0
1
1 1
1
Fig. 1. Labelled directed graph describing the (`= 4, t = 2)
constraint. Taken from [6].
zero otherwise. Let, for example, `= 4 and t= 2, then we
have K= 6 4-bit words of weight t= 2, namely (0,0,1,1),
(0,1,0,1), (0,1,1,0), (1,0,0,1), (1,0,1,0), and (1,1,0,0). The 6×6
transition matrix is [6]
D4,2=
101000
100010
010001
100000
010000
000100
.(1)
Figure 1 shows a graphical description, called labelled directed
graph, of the (4,2) constraint. Hopping from state to state,
following the arrows and reading off the labels attached to the
arrows, produces a (4,2)-constrained sequence.
The (information) capacity of the FSM data source equals
C(`, t) = log2λ, (2)
where λequals the largest (real) root of [12]
det[D`,t zI]=0,(3)
where Idenotes the K×Kidentity matrix and det denotes
the determinant of a matrix. If expanded out, the determinant
is a K-th degree polynomial in z. For the case `= 4 and
t= 2, we find, using (1), that
det[D4,2zI] = z6z5z4z2+ 1 = 0.(4)
Solving the above equation numerically, we find λ1.7143
and C(4,2) = log2λ0.778. Algebraically solving the
determinant (3) is not attractive in case Kis very large as
the storage requirement scales with K2. A slight modification
of the power iteration method [13], on the other hand, finds
the largest eigenvalue of the sparse matrix D`,t with a storage
of order K. More numerical results of C(`, t)can be found
in [6, Table I].
In the next section, we present a simple method for design-
ing (`, t)-constrained codes, named Construction I. The new
construction is valid for all values of `and t, and it does not
require the evaluation of the n-step transition matrix Dn
`,t.
III. BLOCK CODE CONSTRUCTION
In this section, we present a new construction technique for
designing binary (`, t)-constrained block codes that comprise
a single look-up table for encoding user data into a series of
(`, t)-constrained codewords.
The stream of source data is delivered in packets of mbits
that are translated by the encoder, using a single look-up table,
into n-bit codewords, n > m, that are cascaded, serialized,
and transmitted. A concatenation of arbitrary codewords
produces an admissible (`, t)sequence. The decoder
uniquely translates the n-bit codewords, without knowledge
of past or future codewords, into the original msource
words. In our analysis and code construction, it is assumed
that the codeword length nis larger than the window `,n`.
Definition 1 l-constraint: A binary sequence y=
(y1, y2, . . . , yn)satisfies the l-constraint, where lis a
positive integer, l > t, if w(y1, y2, . . . , yl)t, that is the l
leading bits have a weight at least t.
Definition 2 r-constraint: A binary sequence y=
(y1, y2, . . . , yn)satisfies the r-constraint, where ris a
positive integer, r > t, if w(ynr+1, ynr+2 , . . . , yn)t,
that is the rtrailing bits have a weight at least t.
We are now in the position to present the code construction,
where admissible codewords are selected that can be cascaded.
Construction I: Define l=b(`+t)/2cand r=d(`+t)/2e.
Thus, l=rif `+tis even, and l=r1if `+tis odd. Let Sbe
the set of (`, t)-constrained codewords of length n,n`, that
satisfy both the l- and r-constraints. The set of codewords, S,
has the property that no concatenation of codewords violates
the (`, t)channel constraint. The codewords themselves obey,
by definition, the (`, t)constraint, and at the junction of two
codewords the imposed l- and r-constraints guarantee the
(`, t)constraint. If the size of S, denoted by |S|, is at least
2m, we may define a bijective map, {0,1}m{0,1}n,
between the m-bit source words and a subset of size 2mof
the n-bit codewords, S. The map is embodied by a single
look-up table of size 2m.
We experimented with Construction I for a variety of values
of the parameters `, t, m, and n. For small K(not shown),
we found the same codes as presented in the literature [6,
Table III]. For example, for `= 4 and t= 2 we find
2153 (4,2)-constrained 15-bit codewords that satisfy both the
l=r= 3 constraints, so that we can construct a (4,2)-
constrained block code of rate 11/15, see also Section V,
where we shall take a closer look. A survey of block codes
found after applying Construction I, is shown in Table I, where
prior art design methods could not be applied. The parameter
η= (m/n)/C(`, t)defines the rate efficiency of the code.
Note that the capacity C(30,15) could not be evaluated due
to its excessive memory requirements.
A natural question that may arise regards the efficiency of
the new codes constructed. Does, for example, the construction
TABLE I
PARA MET ER S OF BL OC K COD ES F OR A SE LE CTI ON O F `AND tVALU ES .
THE I NT EGE RS mAN D nDE NOT E THE S OU RCE W ORD A ND C ODE WO RD
LE NGT H,RESP ECT IV ELY.K=`
tIS T HE NU MB ER OF S TATES AN D
η= (m/n)/C(`, t)I S THE R ATE EFFI CI ENC Y.
` t K m n η
16 8 12,870 16 20 0.879
17 11 12,376 12 19 0.819
20 15 15,504 13 26 0.792
23 15 490,314 16 25 0.810
26 16 5,311,735 22 31 0.843
28 16 30,421,755 22 29 0.852
30 15 155,117,520 26 31
deliver codes whose efficiency is close to that of a maximal
code or close to capacity?
IV. MAXIMUM-SIZE BLOCK CODE DESIGN
Prior to the assessment of the efficiency of Construction I
in Section V, we discuss transition-matrix-based block code
design. Such a design starts with setting up the n-step K-
state labelled graph with state set Σthat describes the (`, t)
constraint, see Section II. Let the terminal state set ΣΣ
denote a subset of the state set Σ, and let σi, σjΣ, and let
Si,j denote the set of admissible n-bit words that start in σi
and terminate in σj. The set of codewords that emerge from
σiΣand terminate in one of the states σjΣ, denoted
by Fi, is defined by
Fi,[
j:σjΣ
Si,j , i :σiΣ.(5)
Let
SΣ,\
i:σiΣ
Fi(6)
denote the intersection of the |Σ|sets of codewords Fi. The
set SΣconsists of admissible codewords that can start and
terminate in any of the states in Σ. A single look-up table
for encoding and decoding is possible if
|SΣ| ≥ 2m.(7)
If (7) is satisfied, the codewords in SΣcan be uniquely
assigned to the 2msource words. A code with the largest
possible set of codewords for its parameters is called a
maximal code; the maximal code size is denoted by ˆ
N`,t(n).
A maximal code is defined by the optimal state set, denoted
by ˆ
Σ, which is defined by
ˆ
Σ = arg max
Σ
\
i:σiΣ[
j:σjΣ
Si,j
.(8)
Equation (8) involves the generation of sets of n-bit admissible
codewords, Si,j, and computing the union and intersection of
these sets for all possible 2K1state subsets Σ, which is a
colossal operation for increasing `and t.
It was shown by Freiman and Wyner [7], however, that it
is not necessary to consider all possible state subsets Σ.
They showed that it is sufficient to consider state subsets,
called complete terminal sets. Define the partial ordering
’ on the states: that is, σiσj, if every n-bit word
admissible from σiis also admissible from σj, in other words,
if k:σkΣSi,k ⊆ ∪k:σkΣSj,k. A set of states Σ0is a complete
terminal set if σiΣ0,σiσjσjΣ0. The partial
state ordering can be conveniently visualized by a Hasse
diagram, which helps to divide the state set Σinto a number
of smaller complete terminal sets. As a result, the number of
combinations of subsets Σthat has to be considered in finding
the optimal state subset ˆ
Σcan often be significantly reduced.
But, also Freiman and Wyner’s method, although its search
space is much smaller than that of the exhaustive search (8),
rapidly becomes too complex to handle for increasing `and t.
This leaves a designer empty-handed for relevant choices of `
and t.
V. AS SE SS MENT OF CONSTRUCTION I
For the efficiency assessment of Construction I, we exploit
generating functions [14] based on the FSM for enumerating
the number of n-bit binary constrained sequences. Note that
we only need the model for the assessment as it is not required
for the code construction per se.
We define the generating function [15]
G(z),Xgizi,(9)
where zis a dummy variable. Let the operation [zn]G(z)
denote the extraction of the coefficient of znin the formal
power series G(z), that is, define
[zn]G(z),[zn]Xgizi=gn.(10)
Let the elements, hi,j (z), of the K×Kmatrix H(z)denote
the generating function of the number of distinct sequences
of length nstarting in state σiand ending in σj. Using the
transfer-matrix method [14], we obtain
H(z) = (IzD`,t )1.(11)
In other words,
hi,j (z)=∆j,i/,(12)
where ∆ = det[IzD`,t ]and j,i is the j, i-th cofactor of
[IzD`,t ].
The number of n-bit codewords that satisfy the l=b(`+
t)/2c,r=d(`+t)/2e, and (`, t)constraints available with
Construction I, denoted by N`,t(n), can be found by evaluating
the generating function
N`,t(n)=[zn]
q
X
j=1
hq,j (z),(13)
where
q=l
t,(14)
since the Kstates are labelled with the `-bit words of weight
tin lexicographical order, see Section II.
A. Special cases
Below we discuss properties of Construction I-based codes
for special cases of `and t.
1) Case t=`1:The (`, t =`1) constraint is in the
data storage world [16] also known as a minimum runlength
constraint, or d-constraint, where d=tdenotes the minimum
zero runlength. Freiman and Wyner [7] presented maximal
block codes whose codewords start with at least tone’s; only
ntbits of the codeword require (de)coding.
2) Case t= 1:Codes with a maximum runlength of
0’s, or k-constraint, where kdenotes the maximum zero
runlength, have found widespread application in data storage
devices [16]. Note that an (`=k+ 1, t = 1)-constrained
sequence can be obtained by inverting the binary symbols
of a k-constrained sequence. Freiman and Wyner [7] and
Blake [17] showed that a k-constrained block code based on
Construction I is maximal.
3) Case `= 4, t = 2:For the case `= 4 and t= 2, we
find, using (8), that the maximizing state subset equals ˆ
Σ =
{σ1, σ2, σ3}. Codewords in Construction I start with ’011’,
’101’ or ’110’, so that after perusal of Figure 1, we conclude
that the (4,2) constrained code constructed by Construction I
is maximal.
Using (1), (11), (12), (13), and q= 3, we obtain
ˆ
N4,2(n) = N4,2(n)=[zn]
3
X
j=1
h3,j (z)=[zn]Q(z)
P(z),(15)
where
Q(z) = z5+ 2z3z2+ 1,(16)
P(z) = ∆ = det(IzD4,2) = z6z4z2z+ 1.(17)
After an evaluation, we have for n4
ˆ
N4,2(n) = N4,2(n) = [zn4](6 + 10z+ 16z2+ 29z3+
+ 50z4+ 85z5+ 145z6+ 249z7+ 428z8+
+ 733z9+ 1256z10 + 2153z11 +· · · ).(18)
We have ˆ
N4,2(15) = 2153, so that we can construct a maximal
(4,2)-constrained code of rate 11/15 with Construction I. Note
that the same code was also found with Franaszek’s method,
see [6, Table 3]. For larger values of nwe may conveniently
apply the recursion
N4,2(n) = N4,2(n1)+N4,2(n2)+N4,2(n4)N4,2(n6)
(19)
with initial conditions N4,2(0) = 1,N4,2(1) = 1,N4,2(2) = 1,
N4,2(3) = 4,N4,2(4) = 6, and N4,2(5) = 10. For asymptot-
ically large n, the number of sequences grows exponentially
with growth factor λ, or
N`,t(n)A`,t λn, n 1,(20)
where A`,t is a constant. The rate efficiency, η, equals
η=log2N`,t(n)
nC`,t
1 + log2A`,t
nC`,t
.(21)
In case we have a generating function of N`,t(n), say
N`,t(n)=[zn]Q(z)/P (z), then [15]
A`,t =λQ(1)
P0(1).
For the case `= 4 and t= 2, using (3), (16), and (17), we
find A4,20.6636 and λ1.7143, see Section II, so that
η10.76/n.
VI. CONCLUSIONS
We have considered a new design method for construct-
ing energy-harvesting sliding-window (`, t)-constrained block
codes that generate binary sequences with at least t1’s in
a window of `consecutive bits. The block code’s design
avoids the usage of the K-state finite-state machine (FSM)
description of the (`, t)constraint, where K=`
t, so
enabling codes that are hard to design with prior art FSM-
based methods. The block code, using a single look-up table,
translates m-bit user words into n-bit codewords that can
be cascaded without violating the prescribed (`, t)-constraint.
Examples of (`, t)-constrained codes have been presented for
values of `and tthat are too laborious to handle with prior
art design methods. A rate-26/31 (`= 30, t = 15) code could
be established, which, as it requires an FSM description with
more than 1.5×108states, would be practically impossible to
construct with FSM-based methodology. We have assessed the
rate efficiency of the newly developed codes. We have been
able to verify that the (`= 4, t = 2)-constrained block code
designed by Construction I is a maximal block code.
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The use of constrained sequence (CS) codes is important for the robust operation of transmission and data storage systems. While most analysis and development of CS codes has focused on fixedlength codes, recent research has demonstrated advantages of variable-length CS codes. In our design of capacity-approaching variable-length CS codes, the construction of minimal sets is critical. In this paper we propose an approach to construct minimal sets for a variety of constraints based on the finite state machine (FSM) description of constrained sequences. We develop three criteria to select the optimal state of the FSM that enables the design of a single-state encoder which results in the highest maximum possible code rate, and we apply these criteria to several constraints to illustrate the advantages that can be achieved. We then introduce FSM partitions and propose a recursive construction algorithm to establish the minimal set of the specified state. Finally, we present the construction of single-state capacity-approaching variable-length CS codes to show the improved efficiency and reduced implementation complexity that can be achieved compared to CS codes currently in use.
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Bell System Technical Journal, also pp. 623-656 (October)